Everything that I can find says that the transition from the 800 series to the 801 series came late in the year in 1957, so a late 1957 model would be an 8x1. That serial number seems correct too. Jim Smith's "Identifying old Ford Tractors" web page shows the serial numbers for the 1957 800 series being 116368 and up (all 6 digits), while the 801 series from the same year started over at 10001 and went up to 11996 (only 5 digits like yours). The 5 digit serial numbers on the 800 series starting with 115xx were from 1955, not 1957.
I think that you would be safe in assuming that you should order parts specifically for the 801 series, and not the 800 series.